Maggie Gyllenhaal, sitting in a central London office on an inclement Monday, has a stinking cold. The PR person is running about hunting tissues, and I’m gabbling about how she sounds totally normal, when in fact she sounds like a consumptive from a Thomas Hardy novel (and looks like one, too – willowy, pale, intense, troubled, and with a very high neckline).

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Her recent work has been in England and Ireland, and she’s always struck me as a British person’s idea of perfect Hollywood: quirky not polished, indie not mainstream, from a creative film-making family, rather than a dynasty (director father Stephen Gyllenhaal, screenwriter mother Naomi Foner and, of course, brother Jake Gyllenhaal, three years her junior).

So it feels monumentally weird to see the 36-year-old actress on BBC2, in ermine, a crystalline British sensibility, weighed between understatement and profundity. It’s not that she doesn’t pull off her lead role in The Honourable Woman; she inhabits it totally. It’s more that you can’t work out what they did to get her.

Writer/director Hugo Blick (famous for The Shadow Line, Sensitive Skin and loved, by those who remember it, for Marion and Geoff) clearly felt the same, moving heaven and earth (well, dates and trucks) to make it work. “They really made an effort to consider my family, and storyboarded it around the school year. They really were respectful of my life, and I wouldn’t have been able to do it if they hadn’t been.”

She thinks for a moment. She clearly doesn’t want to sound too grateful, if only from a feminist point of view. “This is a project about a woman. It’s written and directed by a man, but it’s a man who is truly interested in women, not a fantasy version of us. It makes sense that he was so considerate of my life.”

She filmed in London and Morocco, with her children Ramona, seven, and Gloria Ray, two, her mother, a nanny… Her husband, Peter Sarsgaard, also an actor, got a job halfway through it, “and I said, ‘I can’t do this if you’re working. I need you.’ For all parents, but particularly if you’re a mother, you go to work all day and then you come home and you do another day. That’s the reality.”

The Honourable Woman is a classic little-screen big moment – with a huge and combustible political backdrop, brilliant plotting, great performances. Gyllenhaal is effusive about the project, says it’s the best thing she’s ever been in, and you don’t get the impression that she’s one of those actors who says that about everything. Besides, she has no reason to be over-excitable – it’s not as though she’s been labouring away in obscurity all this time.

She filmed the controversial Secretary with James Spader, about a sadomasochistic relationship, when she was 22. By the time it came out, cult fantasy drama Donnie Darko was already very significant – even though it was more of a breakthrough for her brother Jake, her supporting role didn’t go unremarked. Of this period, she says, “At 22, most jobs don’t come with a lot of success and power. You’re slowly working up the ladder to get to a place where you have more experience. But with actors, because youth is so valuable… you have all this power. All of a sudden I had this power that I didn’t know what to do with, and I was making lots of mistakes. Well, everybody makes mistakes, but these were very visible.”

Personally, I don’t think the mistakes were at all visible, in either of those films, at least not to the untrained eye. Or perhaps there were flaws that one would never remember, because Secretary, in particular, was such a distinctive way to burst into the culture – risqué, subtle, clever, surprisingly broad in its appeal. I don’t know why I was surprised at the time, except that in my prudish way, I thought sadomasochism would be a hard sell. Then, when it proved so popular, I presumed the universe would put her in a succession of modern-screwball roles that were broadly similar. That’s not how it worked out.

She’s always maintained a deliberate versatility, moving from mainstream to indie to experimental, never slipping back into the same role twice. “There was a time when I was younger when I was given the ‘quirky girl’ part a lot. I find if you do a part and it works and people see it, there’s a time lag where you get offered a lot of things that are like the thing that worked, but meanwhile you’ve grown into something else. And the people who can see that I’ve grown and I’m different to the person I was before, and offer me something different, those are the people I need to be working with.”

One of the interesting things about seeing two siblings, male and female, hit the screen at the same time is seeing the different tightrope they both walk, to do the work they want while not being boxed in by how they are perceived. It’s possible that the box office craves more predictability from Jake, finding him more bankable. But she disagrees. “It’s a testament to my brother how much he’s shifted and changed. Prince of Persia versus whatever he’s doing now… Prisoners. Also that one where he’s a cop? End of Watch! That’s it.”

The film she shot in Ireland was Frank, Jon Ronson’s elegiac sort-of-biopic about Manchester musician Frank Sidebottom. In it, Gyllenhaal played Clara, this incredibly savage, critical beauty. “I went into that movie thinking I was going to be soulmates with Michael Fassbender. And that wasn’t how it happened at all. And then I thought, ‘Wait a second, he’s wearing a big fake head.’ What was it in me, and what was it in Clara, that made me think we were going to make a real connection here?”

So, back to her new role in a drama rooted in Middle Eastern intrigue. “I think this changed me. I read things now and I think, ‘I don’t want to do that.’ The Honourable Woman changed me as an artist and as a person. And things just don’t excite me in the same way anymore.”

Nessa Stein, her character, is the daughter of an Israeli arms dealer who becomes a life peer for her work advancing communications between Israel and Palestine (literally, with cables running data, and figuratively, with the acknowledgment of the need for them). It is an incredibly detailed, layered, profound story, moving like quicksilver through impossible terrain, elegant but determined. My first thought was how controversial this is going to be.

It doesn’t take a stance, but a complex portrait of a protagonist on either side of the Israel/Palestine debate is going to raise hackles. Gyllenhaal has a well-established Democrat pedigree – she’s a supporter of the American Civil Liberties Union and Planned Parenthood. Even though that seems rather mild here, supporting free speech and women’s health, it’s very left-identified, while the LA norm is to be carefully neutral.

She obviously has more stomach for an argument than most, but I still wonder whether she fears the response to The Honourable Woman. “I didn’t feel terrified when we were making it. There were things that terrified me, but not the politics. I think it’s because I really believe that our intentions were… good is a complicated word. What does that really mean? We were after something truly compassionate. This piece doesn’t take a political stance. And I don’t say that out of fear, I say that because it’s true. How can you? It’s an eight-hour long drama, you can’t really have a position over that time.” On such fraught ground, some people will object to the very compassion, though, the fact that it’s not more judgemental. “Well, that’s not my problem,” she says simply.

I’ve only seen one episode of The Honourable Woman, and I already know that, when it ends, it’ll be like losing a relative (an important one; not some random aunt). There is a depth and purpose to the central character that a starlet couldn’t pull off; it’s cometh-the-actor, cometh-the-part.

“I was having a really interesting conversation with [fellow actress] Amy Adams last night about being this age, and what it means, and how different it is to what it used to mean. When you’re 36, you start all of a sudden to go, ‘I’m not 28, I’m 36.’ Take Nessa, for instance: she’s a complicated woman, she’s an emotional woman, she’s a sexual woman, she’s an intelligent woman, she’s a powerhouse. She’s all these things. But she’s not 28. It’s wonderful to have this terrain to step into in my 30s. I did always kind of think that I would have more to do at this age than when I was younger.”